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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005
Convergence and Performance Analysis of
Godard Family and Multimodulus Algorithms
for Blind Equalization
Vinod Sharma, Senior Member, IEEE, and V. Naveen Raj, Student Member, IEEE
Abstract—We obtain the convergence of the Godard family [including the Sato and constant modulus (CM) algorithms] and the
multimodulus algorithms (MMA) in a unified way. Our analysis
also works for CMA fractionally spaced equalizer (FSE). Our assumptions are quite realistic: The channel input can be asymptotically stationary and ergodic, the channel impulse response is finite
and can be stationary and ergodic (this models fading channels),
and the equalizer length is finite. The noise is independent and
identically distributed (i.i.d.). The channel input can be discrete
or continuous. Our approach allows us to approximate the whole
trajectory of the equalizer coefficients. This provides estimates of
the rate of convergence, and the system performance (symbol error
rate) can be evaluated under transience and steady state.
Index Terms—Blind equalization, CMA, convergence analysis,
MMA, performance analysis, Sato algorithm.
I. INTRODUCTION
A
DAPTIVE equalizers have become an integral part of
today’s communication systems. These are used to cancel
the effect of intersymbol interference (ISI) in the channels.
Traditionally, training sequences have been used to estimate
the tap weights in equalizers. However, this consumes some
channel bandwidth. If the channel is time varying, as in cellular
mobile systems, then the training sequence needs to be sent
frequently, and the resulting bandwidth loss can be significant.
In addition, in certain situations, e.g., in point to multipoint
transmission, a training sequence may interrupt with the broadcast if transmitted to revive any disrupted link. Therefore, blind
equalizers have been suggested more recently where instead of
using a training sequence, only general channel input statistics
are used to adapt the equalizer. Some important current applications are in broadband access on copper in Fiber-to-the-curb
and very-high-rate subscriber line (VDSL) networks.
The earliest blind equalization algorithm has been proposed
by Sato [21]. Subsequently, this algorithm was generalized by
Benveniste et al. [3] and Godard [13]. They defined classes of algorithms called BGR (named after the three authors Benveniste,
Goursat, and Ruget) and Godard algorithms, respectively. The
Manuscript received May 26, 2003; revised April 16, 2004. Parts of this paper
were presented at the IEEE International Control Conference, 2003. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Behrouz Farhang-Boroujeny.
V. Sharma is with the Department of Electrical Communication Engineering,
Indian Institute of Science, Bangalore, India (e-mail: vinod@ece.iisc.ernet.in).
V. N. Raj is with the Department of Electrical Communication Engineering,
Indian Institute of Science, Bangalore, India, and also with Philips Consumer
Electronics, Bangalore, India (e-mail: naveen@pal.ece.iisc.ernet.in).
Digital Object Identifier 10.1109/TSP.2005.843725
Constant Modulus Algorithm (CMA), which is one member of
the Godard family, has particularly been widely implemented
because of its simplicity and effectiveness. Since then, other
algorithms have also been proposed: Maximum Likelihood receiver [10], Shalvi and Weinstein [23], Bussgang algorithms [2],
Multimodulus algorithms (MMA) [27], [28], subspace methods
[24], [25], and linear prediction methods [1]. Good recent surveys on the topic can be found in [9], [12], and [24]. However,
CMA continues to be a popular algorithm. Despite extensive research on the analysis of these algorithms, because of the nonlinearities and discontinuities in the cost-functions, further work
is required. In this paper, we concentrate on the Sato algorithm
(because of the historical importance and because it continues to
be a challenge due to discontinuities) and the CMA algorithm.
Since the MMA algorithm, which is an improvement over the
CMA, and the CMA–fractionally spaced equalizer (FSE) can
also be handled in the same way, we have included these in this
paper. In the following, we survey the analytical studies available on these algorithms and then explain our contribution in
this paper.
An early seminal paper on the analysis of the BGR algorithms is Benveniste et al. [3]. They obtained convergence of
the algorithms under the assumptions that the equalizer has a
doubly infinite parametrization and the channel input is continuous and super/sub Gaussian. These assumptions are relaxed in
Ding et al. [8]. They provide the local convergence of the algorithm for an independent identically distributed (i.i.d.) input
to the channel. They also show that under their weaker conditions, there can be an ill-convergence of these algorithms i.e.,
they can converge to an undesired limit point in case of a finite length equalizer. The Sato algorithm has also been studied
in Weerackody et al. [26]. Instead of studying the convergence
of equalizer weights, they study the mean square error of the
equalizer output. They also assume the channel input to be an
i.i.d. sequence, the channel output to be conditionally Gaussian
given the input, and that the equalizer weight vectors are independent of the equalizer input. This approach has been extended
by Cusani and Laurenti [6] to the CMA and by Garth [11] to
the MMA. Macchi and Eweda [14] also analyzed the Sato algorithm when the channel has small ISI. Ding et al. [7] show
that even the Godard algorithm can have convergence to an undesirable equilibrium point. Rupp and Sayed [20] provide the
convergence of stop-and-go variants of the Sato and CMA algorithms under the assumption of no noise and bounded channel
outputs. Yang et al. [27] is an extensive study on the MMA algorithm. We supplement their work by providing the convergence
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SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
of MMA. See the above-mentioned surveys for up-to-date results on this problem.
In this paper, we prove convergence of the Sato and the other
algorithms of the Godard family. Then, we show that our proofs
extend to the CMA fractional spaced equalizers (FSEs) and the
MMA algorithm as well. We show that the trajectory of the
equalizer weight coefficients converges to the solution of an ordinary differential equation (ODE) over any finite time. Due to
constant step size and nonzero noise, even though the ODE may
converge to an attractor, the equalizer will eventually come out
of its neighborhood and may go to another attractor. Our results
hold under more general assumptions than the previous studies:
The input sequence to the channel can be (asymptotically) stationary and ergodic and can have continuous or discrete distributions. This is more realistic than the i.i.d. assumption that is
usually made. Often, the input to a channel is coded (e.g., convolutionally coded), and then, it will be (asymptotically) stationary and ergodic but not i.i.d. In addition, we work directly
on the trajectory of equalizer coefficients than on the convolution of channel and equalizer coefficients, as is often done in
the literature. This has some well-known advantages [9]. We do
not assume that the equalizer weight vectors are independent of
channel output, as assumed in [6], [11], and [26]. In addition, our
convergence proof is explicit and quite general. We can assume
the channel impulse response to be random and time varying.
This allows modeling a fading channel, which one encounters
in wireless networks. The convergence of the Sato algorithm,
due to the discontinuity, has not been rigorously proved so far.
Most convergence results on the other algorithms are also local
convergence results, i.e., when the initial guess of the equalizer
weights is close to the equilibrium point. We do not need such an
assumption. Our method also provides approximate knowledge
of the trajectory of the equalizer coefficients. This tells us about
the rate of convergence of the equalizer weights and, also, as we
will show later, allows us to compute the performance symbol
error rate (SER) of the corresponding system under transience
and steady state. Performance of the CMA under steady state
has also been studied recently in [15], [18], [22], and [29]. They
obtain approximate location of the equalizer under steady state
and obtain bounds on its mean square error. It may be possible
to obtain bounds on SER from these results. However, in our
study, we directly obtain SER not only under steady state but
also under transience.
The paper is organized as follows. In Section II, we specify
our system for the Godard family and also describe our approach. In Section III, we prove the convergence of the Sato algorithm. Section IV proves the convergence for the other members of the Godard family. In Section V, we discuss the performance of these algorithms under transient and steady states.
Section VI describes the setup used for CMA FSE [9] and MMA
[27]. Then, we show how the convergence analysis of CMA provided in Section IV extends to these systems.We also show that
CMA FSE may not have ill convergence, even when there is
some channel noise. Section VII illustrates our results via simulations. In Sections V and VII, we will limit ourselves to the
Sato and CM algorithms. These are the main algorithms of the
Godard family. Of course, in the same way, we can carry out the
details for the other algorithms as well.
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Fig. 1. Communication system model with blind equalizer.
II. SYSTEM DESCRIPTION AND OUR APPROACH
In Section II-A, we describe our system and the equalizer
algorithms we will analyze for the Godard family. Section VI
will describe a modification of this system for the CMA FSE
and MMA. In Section II-B, we present our approach to prove
the convergence of the algorithms.
A. System
We study the following communication system. The input
symbols are passed through the pulse shaping filter
to pro. Let the channel impulse reduce the transmitted signal
.Then, the received signal can be
sponse be denoted by
written as
(1)
Let
denote the impulse response of the cascade of the pulseand the channel response
. If we sample
shaping filter
the received signal at a rate greater than the Nyquist rate, we
can get the sufficient statistics. In this paper, we assume that
this criterion is met. Then, we can have the discrete-time model
where
is a real-valued input sequence that we want to
transmit through the communication channel. The channel is
assumed to be a linear time-invariant system (later on we generalize it to time varying fading channels) with a finite impulse
of order . The output of the channel at time
response
is
. An i.i.d. noise
is added to the channel output
. The resulting system is
at the receiver. Let
shown in Fig. 1. Due to bandwidth limitations, the channel may
exhibit ISI. An equalizer is used at the receiver to remove the
distortion caused by the ISI. We consider only linear equalizers.
To exactly remove the ISI, the transfer function of the equalizer
should be the inverse of the channel transfer function. However,
, and in blind equalization, we also do
we do not know
(no training sequence). Thus, an adaptive
not know the
,
scheme is used to learn the proper equalizer coefficients
is a vector of dimenas the input sequence is transmitted.
sion and
are its components. Let
be
the equalizer output at time . Then,
, where
. Simand
ilarly, we denote
. Often, we will consider a
or
particular realization of . Then, we will denote it by
. We will assume the channel impulse response
and the
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number of equalizer weights to be finite. Based on the input
, the equalizer is updated to
Theorem 1: If the solution of the equation
(6)
(2)
is an appropriate function, depending on the blind
where
equalization algorithm used, and is the step-size constant. In
to the
general, has an effect on the rate of convergence of
steady-state value as well as on the steady-state performance.
We consider the Godard family of algorithms. For this family,
the function in (2) is given by
(3)
, and , an integer, is
.
where
If
, we obtain the Sato algorithm.
provides the
for the Sato algorithm has a
CM algorithm. The function
discontinuity, and hence, it is treated separately in Section III.
The CMA and the other members of the family are considered
in Section IV.
B. Our Approach and Assumptions
For the convergence of the sequence
, we use the results
from Bucklew et al. [4]. For easy reference, we restate the results. The presentation is slightly modified to suit our problem
setup. Define
The following conditions are used in [4].
is asymptotically stationary and ergodic, and
C1)
is an i.i.d. sequence statistically independent
of
. In addition,
is integrable, with
, i.e.,
respect to the distribution of , for each
.
is continuous in
, and for a finite
C2)
is unique for each
as
)
, then under C1 and C2 (for
for any
, where
denotes convergence in proba.
bility, and
We will show that the solutions of the concerned ODEs for
our algorithms do not blow up in a finite time. Then, in fact, in
the statement of the above theorem, we can replace the
and
by .
We will verify the assumptions of the above theorem for the
Godard class of algorithms under the following conditions.
and
are statistically independent.
A1)
is asymptotically stationary and ergodic.
A2)
is a sequence of i.i.d. zero mean random variA3)
ables whose distribution has a bounded probability
density.
for an appropriate
.
A4)
for an appropriate
.
A5)
The and required will be stated whenever needed.
Our assumptions are weaker than those used in any previous
analysis of which we are aware. More importantly, they will be
satisfied in most practical systems. Actually, the proof of convergence can be used under the more general conditions when
is random instead of being
the channel impulse response
is stationary erdeterministic. Then, we will assume that
godic and independent of
. This implies that
is also asymptotically stationary and ergodic ( denotes
the convolution operator). In addition, in that case, we will need
.
We will use the above result on the Sato algorithm in Section III and on the rest of the Godard class in Section IV.
III. CONVERGENCE OF SATO ALGORITHM
(4)
for every
The Sato algorithm can be described as
and
sgn
(7)
(5)
, and sgn
where
Under the above conditions, the following result is proved in
[4]. We use the notation
, where
denotes the
integer part of . Define
for , which is a suitably large constant. Then, we have the
following theorem.
sgn
for
for
is defined as
.
The sgn function makes
a discontinuous function,
and hence, this algorithm has been considered harder to analyze
as compared to some other blind equalization algorithms. However, we will see that our convergence results will go through,
SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
despite this discontinuity. This happens because in the assumpto be continuous only at the tration C2, we require
and
corresponding to
jectory of ODE (6). Define
, as in the begining of Section II-B.
Now, we verify [4, assumptions C1 and C2] mentioned in
is assumed
Section II. Since the channel impulse response
is asymptotically stationary and
to be of finite duration, if
ergodic, then
, being a convolution of
with
, is
also asymptotically stationary and ergodic. In addition, from (7),
,
if
for any given
. Therefore, condition C1 is verified.
Next, consider condition C2. If
, then (4) is sat. Of
isfied. Similarly, (5) is satisfied if, in addition,
if
. Now, consider the conticourse,
. It is provided by the following Lemma.
nuity of
,
Lemma 1: Under assumptions A1–A5 with
is continuous in
with probability one.
Proof: From (7)
sgn
sgn
(8)
Thus, we need to show the continuity of
sgn
and
sgn
with respect to
.
sgn
. The funcFirst, consider
is continuous, except at zero. Therefore, when
tion sgn
, sgn
is continuous in
if the
argument is not zero. If the distribution of has an absolutely
continuous component, then this happens with probability zero
. Similarly, one can say when
.
for any given
Therefore, when
, if
, then
sgn
sgn
almost
, this implies,
surely. Since we have also assumed
by the dominated convergence theorem, that
sgn
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. This shows that the ODE has a unique local solution
at
. Then, we argue that
for any initial condition
does not matter. Finally, in Lemma 3, we prove our assertion of
.
unique solution for all time if the initial condition
,
Lemma 2: If A1–A5 are satisfied with
is locally Lipschitz for
.
Proof: See the Appendix .
is certainly not a desirable equalizer for any
Since
channel, for any reasonable equalizer algorithm, zero should not
be an attractor. This is also true for the Godard family. At least
for the Sato and CM algorithms, one can observe from the cost
functions [see (21) and (22) below] that they minimize. An equilibrium point is an attractor for the ODE if and only if it is a local
minimum of the corresponding cost function. Zero is not a local
minimum for these cost functions. The example in Section VI
also shows it. When has a density (assumption A.3),
with probability one, and if zero is also not an attractor, it can be
,
with probability one
ignored (because even if
), and we can start the ODE with
. Thus,
for
in the following lemma, the restriction of
is harmless.
,
Lemma 3: Under the assumptions of Lemma 2, if
the ODE has a unique solution, and the solution does not blow
up in finite time.
Proof: To prove our assertion, we use a result from [17, pp.
is continuously differentiable
171]. This shows that if
(
denotes a norm) by
and we can upper bound
, i.e.,
a Lipschitz function of
for each , then the solution of our ODE will not blow up in a
finite time. Since
sgn
define the function
as
Then
sgn
(10)
However, one can show that
sgn
is, in
,
general, not continuous under our assumptions at
but when has an absolutely continuous component, from (7),
with probability zero for
.
we observe that
Now, consider the continuity of
sgn
.
, the argument of
Since this also involves sgn
the last paragraph works for this also. This gives the continuity
on a set of
with probability one.
of
Next, we consider the ODE
(9)
needed in Theorem 1. We would like to show that (9) has a
unique solution for each initial condition and that the solution
does not blow up in finite time. This would hold if we could
is Lipschitz. However, it is easy to show that
show that
it is not. Therefore, we split the proof of our result in two parts.
is locally Lipschitz
In the next lemma, we show that
and hence, is Lipschitz. This proves (along with the fact that
, there is a unique solution
0 is not an attractor) that if
of the ODE, and the solution does not blow up in finite time.
Using the above results, we have proved the following.
Theorem 2: For the Sato algorithm, under the assumptions
, and
A1–A5, with
(11)
, for any
as
the ODE
. Here,
is the solution of
(12)
with the initial condition that
.
From the above theorem, we can show the following useful
be an attractor of the ODE (12) and
be a point
result. Let
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such that if
, then
from (11), we obtain that as
, as
, for any given
. Then,
for
and
large enough. Thus, for small enough ,
is concentrated
asymptotically (in time), the trajectory of
with a large probability for a long time. For all pracaround
can be taken as its
tical purposes, for small enough , then
will eventually come out of
steady-state value, although
,
its neighborhood. A similar conclusion is made if instead of
of the ODE (12), which has a rewe have an invariant set
gion of attraction. If
is not an equilibrium point but is a renor
will converge to
unless
peller, then neither
(then,
stays there, but
will move out of
has a region of attraction but does
it and will not return). If
not contain a ball with
as its center, (i.e.,
is a saddle
may converge to it, but
will not, at least
point), then
when there is noise. Similar conclusions hold for the convergence results in Sections IV and VI.
From (11), we also obtain an approximation of the trajectory
for small . We will use it to obtain the performance
of
of the system at any time in Section V. To be able to obtain the
.
above useful information, we need to compute
One can easily show that
with
Proof: We first show the continuity of
. Let us denote
in (3) as a product
respect to
(when
are functions of , , and
representing the three terms in the three square brackets in (3)).
be a sequence converging component-wise to
Let
. For large enough ,
for appropriate finite and . Then
and
Therefore, for large enough
(13)
is the distribution function for ,
, and
is the probability
density of . We will use these simplified expressions on an
example in Section V.
(14)
Expectation of the right side of (14) is finite if
. Then, by continuity of
(and hence of
) as a
, by the dominated convergence theorem
function of
where
,
IV. CONVERGENCE OF GODARD FAMILY
) OF ALGORITHMS
(
In this section, we prove the convergence of the Godard
, defined in (3) in Section II.
family of algorithms, for
Define
We assume (A.1)—(A.5) in Section II, with and to be
specified. In the following, we prove C1 and C2 of Section II and
is locally Lipschitz and that the solution
also show that
of the ODE does not blow up in finite time.
is asymptotically staWe already know that
tionary and ergodic. One can also easily check that
for any
whenever
. This verifies C1.
The following lemma verifies C2.
, C2
Lemma 4: Under assumptions A1–A5 with
is satisfied.
and
is finite for all
. Under the same con.
ditions, we also get
, we also need
For
. Thus, condition C2 is verified.
Next, we consider the ODE
(15)
To ensure the existence and uniqueness of a local solution
to (15), we need to verify the local Lipschitz continuity of
. We do this in the following Lemma.
is locally Lipschitz under the assumpLemma 5:
tions of Lemma 4.
. Later on, we
Proof: We first prove the result of
.
will have a proof for
By arguing as in Section III, it is enough to verify that
is continuously differentiable with respect to .
For this purpose, we express
as
, as
is
discussed in the proof of Lemma 4. We observe that
obviously differentiable. For differentiability of and with
respect to , we only need to verify the differentiability of
for : a positive integer. This, in fact, will be
SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
verified if we verify it for
. One can easily check its
. Under our
differentiability with respect to whenever
occurs only with probability zero. For
assumption A3,
convenience, we write the derivative of
(we will write
) for
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. Then, certainly,
Define
as
. In addition
sgn
(17)
(16)
, the middle term in the above expression is absent.
When
and
,
Thus, under assumptions
) of the derivative of
expectation (with respect to
with respect to
can be upper bounded by a function that is
. Therefore, by the
absolutely integrable with respect to
dominant convergence theorem, we can take the gradient operinside the integral,
ator in the expression
. In addiand hence, we obtain the differentiability of
with probtion, the derivative in (16) is continuous for
ability one. Again, by the dominated convergence theorem, we
for
.
obtain the continuity of derivative of
. Consider a
Now, we show Lipschitz continuity for
of center
and radius
. Observe that
ball
for
. Thus, for
large enough, the term in the
We need to show that for all
bracket in the last equality is positive.
Define
(18)
By the dominated convergence theorem, the map
is continuous on the unit circle, and hence, the
infimum in (18) is finite. In addition, under the assumption
for any on the unit
that has a density,
circle for any
. Therefore, since infimum of a continuous
.
function is attained on a compact set,
The term in the bracket in (17) can be written as
(19)
. Therefore, (19) is positive whenever
. This proves the lemma.
Finally, we obtain the following.
Theorem 3: For the Godard family of algorithms (with
) characterized by the function
in (3), if A1–A5 is satisfied with
and
, then for any
where
as
Therefore, when
and
,
is locally Lipschitz at
.
As in the case of Sato, one can observe from (16) that the
is not bounded. Therefore, we have only
derivative of
is locally Lipschitz, but the global Lipsshown that
chitz property is not guaranteed. This is not sufficient to guarof the ODE (15) will not blow up in
antee that the solution
a finite time. However, now, we use another technique to guarantee this.
Lemma 6: Under the assumptions A1–A5 with
, the ODE (15) has a unique solution for any given initial
condition, and the solution does not blow up in a finite time.
Proof: To prove this result, we use the following fact. If
the ODE has a unique local solution and there is a continuous
, such that
as
function
and
whenever
is large enough, then the
solution of the ODE does not blow up in a finite time. Below,
we show such a function.
. Here,
is the solution of the ODE
As in Theorem 2, here also we have proved that the trajectory
spends a lot of time around an attractor
with a large
of
if
as
.
probability as
Next, we give a simplified form of
(see details
is i.i.d.
[
denotes
in [16]) when noise
normal distribution with mean 0 and variance ]. We will use
.
these simplifications in Section VII to plot the solution
Denoting
(20)
where
.
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V. PERFORMANCE ANALYSIS OF SATO AND CM ALGORITHMS
The performance of the communication system described in
Section II at any time depends on the values of the equalizer
weights at that time. Starting from any point, the trajectory of
the equalizer coefficients is given by the process
. This process, by Theorems 2 and 3, is approximated by the
of the corresponding ODE. If the ODE
solution
converges to an attractor, the steady-state performance of the
system can be approximately obtained by taking the equalizer
weights at the attractor. Thus, in this section, first, we study the
attractors of the ODE and then look at the overall trajectory to
obtain the system performance under transience also.
In Section V-A, we study the attractors of the algorithms. In
Section V-B, for a particular system, we compute the symbol
error rate for a given value of the equalizer. This allows us to
compute the performance at any time. We will illustrate it by an
example in Section VI.
A. Attractors of the ODEs
Similarly, defining
we get, when
is i.i.d.,
(24)
and
diag
The cost functions for the Sato and CM algorithms are
sgn
(21)
(22)
. The derivative of these functions form the
where
right side of the ODE to which these algorithms converge (as
). Thus, the equilibrium points of the ODE are the points
, where the derivative vanishes, i.e., when
.
We have established earlier that the algorithms can converge
to equilibrium points, which are attractors. Attractors are the
equilibrium points that are local minima of the cost functions.
When the cost surface has multiple minima, then we may
have ill-convergence, i.e., convergence to local minima that
are not global minima. Since ill convergence can degrade the
performance of the system substantially, it has been studied
extensively (see the surveys [9], [12], and [24] for the noiseless
case, and [18] and [29] for the noisy case). In particular, for
finite length equalizers, there can be ill convergence. However,
all these results are for i.i.d. channel input. Under our general conditions, the situation can be more complicated. The
attractors and their regions of attractors depend on the input
and noise distributions, but the possibility of ill-convergence
persists. For fractionally spaced CMA, which will be discussed
in Section VI, the ill convergence can be avoided under the
small noise variance.
Let us first consider the Sato algorithm. In general,
will have multiple zeros. If the Hessian of
is positive definite at a zero of
, then it
is a local minimum of the cost function and an attractor for the
is i.i.d., with distribution
ODE. We can show that if the noise
, then
(25)
represents a diagonal matrix with the th elewhere diag
ment given by . We will use (23) and (25) to obtain the equilibrium points (the zeros of
) for the Sato and CM algorithms for a specific communication system we describe below.
From the equilibrium points, we will find the attractors. If one
also knows the regions of attraction of these attractors, one can
predict to which attractor the ODE will converge for a given
will initially tend to that
initial condition. The process
attractor and stay around it for some time and then again wander
) or even diverge. If is
away to another attractor (since
very small, the time spent around an attractor can be very large.
The following lemma is used in subsequent sections.
is i.i.d.
, the attractors
Lemma 7: When noise
of CMA depend continuously on noise variance , and there is
no bifurcation in the branch of attractors with respect to noise
variance .
is a
Proof: From (24), we observe that
. In addition, from (25), the Hescontinuous function of
sian of
is a continuous function of
in the open set
. Therefore,
is continuously differentiable
.
in
of
restricted to the
The total derivative
affine subspace {
and
fixed} is the
. It is positive definite at an atsame as the Hessian of
tractor of CMA and, hence, invertible.
represent an attractor corresponding to noise variance
Let
. Then, by the implicit function theorem [19], there exist open
and of and a mapping
from
neighborhoods of
to such that
and
for all
. In addition, by continuity of the Hessian of
as a
function of , the Hessian of
at these solutions stays positive for small . Hence, these solutions are attractors. Further,
due to the implicit function theorem, there is no bifurcation at
[5, p. 129].
the attractor
B. Performance Analysis
(23)
Given the equalizer coefficients, one can study the performance of a communication system. As an example, we
SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
consider a particular pulse amplitude modulation (PAM)
system and compute its symbol error rate (SER) at particular equalizer coefficients. Using these, we will obtain the
system performance under transience and steady state. We
to be i.i.d.
. For
will take the noise sequence
,
, define
Thus, the equalizer output, with input
, is
(26)
is the random variable reprewhere
senting the distortion due to the ISI in the th output of the
equalizer. The PAM constellation used is shown in Fig. 2, where
and
is the energy in the pulse used to transmit the symbols.
is the parameter used to control the spacing between the signal
points and, hence, the average signal power of the constellation.
To compute the probability of a symbol error, we need to find
the probability that the summation of the noise term and the
ISI term exceeds in magnitude one half of the distance between
.
two levels. In our case,
After the transmission of the PAM signal through the channel
and the equalizer, the minimum intersymbol distance reduces
. Thus, the minimum intersymbol distance in
by a factor of
is somehow
the new constellation is (we are assuming that
available at the receiver)
(27)
Then, the probability of error at the receiver is
error when
For
is transmitted
i.i.d., this can be simplified to
(28)
.
where
Similarly, one can obtain expressions for other constellations.
We can use the above expressions for SER and the ODE ap-
Fig. 2.
1527
PAM constellation.
proximation of the equalizer trajectory for the Sato and the CM
algorithm to compute the trajectory of the SER for the two algorithms. We will compare the trajectory so obtained with the
simulation results in the next section.
VI. FURTHER EXTENSIONS AND GENERALIZATIONS
In this section, we extend the results provided so far to other
systems. First, we consider in Section VI-A the CMA FSE or,
equivalently, the CMA single input multiple output (SIMO)
equalizer. In Section VI-B, we consider the recently proposed
algorithm MMA. The convergence proofs for these equalizers
require minor modifications to the proofs in Section IV. Therefore, after explaining these systems, we will only make the
necessary comments.
A. CMA FSE
Let the continuous time channel output of the system in Fig. 1
be sampled at times the baud rate (
). If
, then
, then it
we obtain the system studied so far, but when
is an oversampled system. This can lead to improvements in
performance. For example, if a length-and-zero condition [9] is
satisfied for a certain upsampling factor , then taking sampling
rate times the baud rate will ensure that the FSE CMA will
no longer have ill convergence under noiseless conditions. In
addition, the required equalizer length can be small.
, the sampled
In the case of oversampling at rate times,
channel output can be subdivided into subsequences, each of
which is the output of a discrete linear time invariant channel
. Each of these channels can be equalized by a
with input
separate CMA equalizer. Such a system (see Fig. 3) is called an
FSE CMA.
Fig. 3 can also represent an SIMO channel where there is a
single input sequence
, and the receiver gets the output
from separate channels (e.g., multipaths in a fading wireless channel or the receiver with multiple antennas). Thus, this
system is also covered by our analysis.
is the
The following notations are used in this section:
th component of th equalizer at time , and
is the noise in
th subchannel at time . The outputs of the channel (with and
without noise)
and
are represented by similar notation.
is the th component of the th channel.
The output of the equalizer can be written as
with
, where
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005
Fig. 3. Block diagram of an FSE.
Further,
, which is the channel output, can be expressed in
and the channel co-efficients
terms of the input sequence
as
, where
Fig. 4. Communication system model (using CAP architecture) with blind
equalizer.
and
B. MMA
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
..
.
The CMA FSE equalizer is updated as
The only difference between CMA and CMA FSE is in the
is related to input. Now,
. It is easy to see
way
is asymptotically stationary ergodic, then so is
.
that if
With the above mentioned notational changes, the proof of a
theorem corresponding to Theorem 3 for convergence of CMA
FSE is same as for CMA.
Next, we consider the attractors. It is shown in [9, ch. 7] that
under i.i.d. input conditions and zero noise, if the length and
and no two chanzero condition (equalizer length
nels have common zeros) is satisfied by the channel, CMA FSE
has only one minimum in each hyper cone
for all
, where
, and
is the th component of . Each of these minima is a global minimum.
One can show that Lemma 7 holds for CMA FSE. Thus, there
is no bifurcation in any branch of attractors with respect to noise
variance. This implies that for any noise variance , the number
of attractors remains the same. Hence, by continuity of attractors
as a function of , for very small noise variances, each cone
still contains a unique attractor. These attractors will be close
in their cones.
to the global optima corresponding to
The performance of the system SER depends continuously on
the equalizer coefficients, as is evident from (28). Therefore, for
, the SER at limit points will stay close to that of
small
the globally optimum equalizer.
In this section, we study the MMA, as described in [27]. The
communication system considered for this algorithm in [27] is
the Carrierless Amplitude and Phase (CAP) modulation transceiver. In the following, we first briefly describe this system.
Then, we will explain the MMA algorithm for this system. This
algorithm can perform better than CMA for nonsquare and
dense constellations. However, we will see that convergence of
this algorithm is an easy extension of the convergence proofs
in Section IV.
We study the communication system in Fig. 4. Now, the
is complex valued with
.
channel input
In the CAP system, the role of carrier to preserve the phase
information is taken care of by two digital filters with impulse
and
modulating the real and imaginary
responses
part separately. These filters operate at a rate greater than
), where is the symbol interval. The
and
are
(
orthogonal to each other and have unit energy. Therefore, we
choose a Hilbert pair for this purpose.
and . The
The outputs of the upsampler are denoted by
upsampling is done by adding the necessary number of zeros
rather than using interpolation. The higher sampling rate is
used only to ensure orthogonality at the receiver and, hence,
to demodulate the channel real and imaginary outputs independently. This requires the equalizers to be initialized with
corresponding sequences of a Hilbert pair, as will be explained
, output , and noise
later. The channel impulse response
are as in Section II. In addition,
. To
exactly remove the ISI, the transfer function of the equalizer
should be the inverse of the channel transfer function convolved
with the corresponding waveform of the Hilbert pair. An adaptive scheme is used to learn the proper equalizer coefficients
and
, as the input sequence is transmitted.
and
, which are the values of the equalizer coefficients at time
, are vectors of dimension
with respective components
and
.
be the output of equalizer , and let
be the output
Let
of equalizer
at time . Then,
, where
. Similarly,
. We denote
SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
and
the equalizer is updated as
. Based on the input
1529
,
(29)
where
for MMA is given by
(30)
and
.
and
are downsampled to the
The equalizer outputs
symbol rate. This sequence is fed to a nearest neighbor decision
and
of input symbols
device to obtain the estimates
and .
These equations individually are similar to the equations for
CMA analyzed in Section IV, except for the upsampling at the
transmitter. Thus, making the notational modifications, as in
Section VI-A, we can obtain convergence of their trajectories to
the corresponding ODEs. Equations (29) and (30) can be studied
independently, and their attractors can be obtained as for the
CMA algorithm in Section V. In Section VII, we will demonstrate the convergence of the trajectories to the corresponding
ODEs via simulations. Once we have the equalizer coefficients
at any given time, we can study the system performance at that
time.
One can also use a CMA algorithm as an equalizer for this
system (use details in [27]). Then, of course, the convergence of
this algorithms holds, as in Section IV.
VII. SIMULATION RESULTS
In this section, we apply the theoretical results obtained and
verify them via simulations for a particular PAM communication system for the Sato and the CM algorithms. For the MMA,
we consider a CAP system at the end of the section.
Let us consider the channel with impulse response
for
. Let the input symbols
be i.i.d. with values
. The equalizer has two taps.
in
is i.i.d.
.
The noise
One can show that for the above channel, the Sato cost func), ( , 0.3), (0,
tion with no noise has local minima at (1,
). Since the cost function of Sato (and also of
0.89), (0,
CMA) is even, we need to study only two of these minima. The
at (1,
) and (0,
) is
with the corresponding eigenvalues {0.7791, 1.4382} and
{0.7792, 1.4386}, respectively. This shows that all the equilibrium points are local attractors. Similarly, we can show that
for the no-noise case, the cost function of the CMA has zero
Fig. 5. SAT-scaled trajectory and Sato ODE with step size
channel 1 with different initial conditions.
gradient locations (equilibrium points) at (1,
( , 0.3), (0,
), and (0, 0). The
= 0:001 for
), (0, 0.888),
is
The eigen values of these matrices are {3.5139, 1.3949},
,
}, respectively. This
{3.9435, 0.8576}, and {
shows that all the equilibrium points except (0, 0) are local
attractors.
The simulation is done for different values of , different initial settings for the equalizer taps, different values for SNR, and
different source distributions. Thereby, our analysis explains the
behavior of the Sato and CM algorithms in transience and steady
state for a wide range of possible variations in the system settings.
and
for the Sato algorithm for
In Fig. 5, we plot
different initial conditions.. The different parameters are
, noise variance
, and the source distribution is uniand
overlap almost perfectly. The plots
form. The
) and (1, 0). For the
correspond to initial conditions (0.5,
),
given channel, the optimum equalizer coefficients are (1,
but the trajectory corresponding to the initial condition (0.5, )
), which is not an optimum
converges to the limit point (0,
point. This illustrates ill convergence. However, when started
from (1,0), the algorithm converges to the optimum equalizer
coefficients. This establishes the fact that there exist multiple
equilibria. The trajectories differ in their limit values, depending
on the initial conditions.
Fig. 6 plots the curves for the same channel and parameters
), (1, 0), and (2,
for the CMA with initial conditions (0.5,
2). All above observations continue to hold, except that the
rate of convergence is much faster, and the oscillations are more
in CMA.
1530
Fig. 6. CMA-scaled trajectory and CMA ODE with step size = 0:001 for
channel 1 with different initial conditions.
Fig. 7. Sato-scaled trajectory and Sato ODE for channel 1 with different values
for SNR = 30, 20, and 10 dB.
We have also simulated the trajectories for Sato and CMA
for different values of (0.001 and 0.005) with parameters
dB and the input distribution uniform. The results
SNR
are provided in [16]. Due to lack of space, we will not report
them here. From this, we can infer that even with noise, the ODE
. As increases, the oscillations increase. As
tracks the
increases, the rate of convergence also increases.
In Fig. 7, we have shown the behavior of the algorithm
and the solution of the ODE
for different values of SNR
(which is also the noise power in this case since the signal power
is 0 dB). From this, we can infer that the noise level can change
increase with noise.
the limit points. The oscillations in
very well as the noise
The theoretical ODE does not track
level increases (at 10 dB). Fig. 8 shows the corresponding results
for CMA with the same qualitative conclusions.
In Fig. 9, we have plotted the trajectories for different source
distributions. The trajectories in the figure correspond to the uniform distribution and to the distribution (0.4, 0.4, 0.1, 0.1) for
. From this, we infer that the change in
the input symbol
IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005
Fig. 8. CMA-scaled trajectory and CMA ODE for channel 1 with different
values of SNR = 30, 20, and 10 dB.
Fig. 9. Sato-scaled trajectory and Sato ODE for channel 1 with different source
distributions with 20-dB SNR.
Fig. 10. CMA-scaled trajectory and CMA ODE for channel 1 with different
source distributions with 20-dB SNR.
input distributions can change the attractors and the ODE trajectory. One draws the same inferences for CMA from Fig. 10.
SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
Fig. 11. SER evolution for the Sato equalizer with 15-dB SNR and nonuniform
source distribution (0.4, 0.4, 0.1, 0.1).
Fig. 13. MMA trajectory and MMA ODE with step-size
channel 1 with 20-dB SNR.
1531
= 0:001
for
We have also done simulations for a channel with impulse
, 0.2572, 0.1221,
, 0.1167,
response (1.0000,
,
, 0.0356,
,
). The optimum
three-tap equalizer has the values (1,0.815,0.407). We found
similar conclusions for Sato as well as CMA. The results are
provided in [16].
Finally, we provide simulations for an MMA system. In the
simulations, we have used an 8-QAM constellation given by
.
}. We
The channel has the impulse response {1, 0.3,
upsample the encoder output by a factor of 4. The new sample
values are zeros embedded in the intermediate sample instants.
,
] and [
,
We choose a Hilbert pair [0.5,
] in our simulations. Gaussian noise is used to
0.5, 0.5,
corrupt the output of the channel. The initial impulse response
of the equalizers are the same as their corresponding Hilbert
sequences. The MMA algorithm is used to update the equalizer
and the coefficients. The ODEs and the simulated values are
and SNR 20 dB.
shown in Fig. 13 for
Fig. 12. SER evolution for the CM equalizer with 15-dB SNR and nonuniform
source distribution (0.4, 0.4, 0.1, 0.1).
We also observe from these that the limit points are different in
both the algorithms, in spite of maintaining all the parameters
the same. It is also seen that the Sato algorithm has higher rate
of convergence than that of a CM algorithm, when the source
distribution is (0.4, 0.4, 0.1, 0.1). The rate of convergence observation here contradicts the one made above for uniform input
distribution. Thus, one cannot conclude that one algorithm converges faster than the other in all conditions.
The SER trajectory for this example is plotted for the two
algorithms in Figs. 11 and 12. The simulated SER is obtained
by running the simulation for a long time at the given equalizer
values. We provide the plots for 20 dB SNR with source probability mass function (0.4, 0.4, 0.1, 0.1). The figures provide the
theoretical curves obtained above as well as the results obtained
via simulations (for the simulated trajectory). One sees a close
match between theory and simulations in all cases.
APPENDIX
PROOF OF LEMMA 2
is continuously differIt is sufficient to show that
. We will show it for the component
of
entiable for
. Similarly, one can be shown for other components of .
, if we show that
Since
exists for a fixed
and all and
, where
are the component functions of
and
are integrable, then by the dominated convergence
theorem,
is differentiable with respect to
and
. Furthermore, again by the dominated convergence theorem,
is continuous if
is
continuous for
at all . Thus, we prove these facts for
for any
and any .
From (7), we observe that we only need to consider the funcsgn
and
sgn
.
tions
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IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 53, NO. 4, APRIL 2005
First, consider
sgn
sgn
. For a given
Next, consider the case when
. Then, (32) equals
implies
. Since
, this
(31)
with
Thus, we only need to show that the derivative of
exists and is continuous with respect to
when
respect to
. For a fixed
and , since
has a density,
also has a density, and hence,
.
is
We have (32), shown at the bottom of the page, where
the density of .
by
. We consider (32), when
Denote
and when
separately. First, consider
. Because
has a density,
also has
when
a density, which we denote by
. Then, (32) becomes
, we
(33), shown at the bottom of the page. For a fixed
of
is bounded (say
show below that the density
), and hence, the integrand in the above integral is
by
. Since
has been
bounded by
assumed, by the dominated convergence theorem, the limit as
can be taken inside the integral, and hence, (33) equals
(34)
Now, we show that
is continuous in and bounded
. We know that
is a convolution
whenever
when
and
. Since
,
of densities of
for some
. Then, density
of
is given
by
(37)
Using (36) and (37), we have
Since
and
, the right side of the above
inequality is integrable.
Next, we show the continuity of the derivative of
. From (35), we observe that for
,
is
a continuous function of . We have also shown above that for
is bounded. Thus, by the dominated converis continuous in for
. Again,
gence theorem,
using (34), by the dominant convergence theorem and from (37),
we obtain the continuity of the derivative of
with respect to .
sgn
Next, consider the differentiability of
with respect to . Consider the th component of the
right side of the equality
sgn
(38)
where
represents the indicator function. Since by assumption A3 the components of are i.i.d., we obtain
(39)
(35)
is an upper bound on the density
of the noise
where
. Since the density of
is a convolution of densities
for
with
, and the convolution of densities
of
is upper bounded by the upper bound on any one of them, we
. Thus,
obtain
from (34)
(36)
when
.
We consider the derivative of it with respect to the th
of . Using arguments similar to showing
component
sgn
and the
the differentiability of
dominated convergence theorem, we have the deriva, which equals (assuming
tive of
), (40), shown at the top of the next page, where
is defined as
.
the random variable
The finiteness of the expression in case of
is obvious from
.
an argument given above with the assumption
, we can show, in the same
For the expression in (40), for
of
is bounded by
way, that the density
for
. Hence, under the assumption
,
(32)
(33)
SHARMA AND RAJ: CONVERGENCE AND PERFORMANCE ANALYSIS OF GODARD FAMILY AND MULTI MODULUS ALGORITHMS
1533
for
for
(40)
and
are bounded, the
the integral is finite. Since
two integrals in (40) can be bounded by integrable functions
. Using arguments similar to showing
of when
sgn
is continuous with respect to
that
, we can show that
sgn
is also
continuous with respect to .
ACKNOWLEDGMENT
Various discussions with V. Borkar and V. Kavitha have been
very useful.
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Vinod Sharma (SM’00) received the B.Tech. degree
in electrical engineering from Indian Institute of
Technology, Delhi, in 1978 and the Ph.D. degree in
electrical and computer engineeering from Carnegie
Mellon University, Pittsburgh, PA, in 1984.
He was an Assistant Professor with Northeastern
University, Boston, MA, from 1984 to 1985 and a
visiting faculy member with the University of California, Los Angeles, from 1985 to 1987. He has been
a faculty member with the Indian Institute of Science, Bangalore, since 1988, where he is currently a
Professor with the Electrical Communication Engineering Department. His research interests are in modeling, analysis, and control of wireline and wireless
networks. Recently, has also been interested in information theory and signal
processing aspects of wireless channels.
V. Naveen Raj (S’02) received the B.E. degree in
electronics and communicatioon from Anna University, Chenai, India, in 1998 and the M.Sc.(Engg.) degree in communication engineering from Indian Institute of Science, Bangalore, India, in 2002.
He is currently with Philips Consumer Electronics,
Bangalore, where he is involved in the development
of audio codecs. His research interests are in the areas
of statistical signal processing algorithms and information theory applied to communications and audio
compression.